13 September 2007

the Bananach-Tarski paradox

From Courtney Gibbons' Brown Sharpie: the Bananach-Tarski theorem. This tells us that a banana can, through some sort of magic, be made into two bananas.

This is of course a restatement of the Banach-Tarski theorem, which tells us that a solid ball can be partitioned into finitely many non-overlapping subsets, which can then be rearranged to form two balls each of the same size as the original ball.

Of course, you couldn't do this with actual bananas (even though a topologist can't tell a banana from an orange, or a doughnut from a coffee cup, or eir pants from a hole in the ground). Damn atoms.

What other fruit-related math jokes are there? The only one I can think of is "What's purple and commutes? An abelian grape," which barely qualifies as a joke.

I found the Bananach-Tarski paradox via the Facebook group I support the right to choose one element from each set in a collection, which asks:
Do you believe that an infinite product of nonempty sets should be nonempty? Do you feel that non-measurable subsets of the reals should exist? Or games of perfect information with no winning strategy for either player? Do you believe that every set should have a well-ordering, or that any poset in which every chain has an upper bound is entitled to a maximal element? Do nonzero rings have the right to a maximal ideal? Is every vector space entitled to a basis? Should fields have algebraic closures? Should products of compact topological spaces be compact, and countable unions of countable sets be countable? Do you want to be able to cut a sphere up into a finite number of pieces and reassemble them, with only rigid motions, into a sphere twice as large?


Finally, because this blog is at least nominally about probability: imagine a (randomized) algorithm that outputs a string of letters as follows: start with any three letters a1, a2, a3. Then for each n > 3, consider some corpus of English text, and consider all the occurrences of the string an-3 an-2 an-1 in that corpus. Tabulate the distribution of the letter that follows that string in the corpus. Choose an from that distribution.

I don't have the programming skills or the corpus to work from, but I bet there's a significant chance that if you start with BAN you get BANANANANANA...

49 comments:

Anonymous said...

This reminds me of my favorite category, the category of Banach Analytic Manifolds. This is because, if you follow the usual abbreviation of using the first three letters of each word, you get:

BanAnaMan

Its not really related, but how often are you discussing banana-related math puns?

Michael Lugo said...

greg,

The real question is: are there any banana-related math puns that don't involve Banach?

Jadagul said...

My apologies for the following, but you did ask. I have a friend who sort of collects these.

Q: So what do you call something that's small, purple, and commutes, and is worshipped every other evening?

A: A binightly venerated abelian grape.

Q: What's yellow, normed, and complete?

A: a Bananach space.

Q: Why did the chicken cross the Mobius strip?

A: To get to the other...the other...um...(damnit!)

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I thought the answer to the last one was "to get to the same side".

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A: The lime at infinity.

Q: What's yellow and equivalent to the Axiom of Choice?

A: Zorn's Lemon.

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A: The irrelephant ideal.

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Have you heard what I believe is the best math joke ever?

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